We observe that \(X^{(\lambda _1,\lambda _2)}\)  is a null vector (a vector in the light cone \(\mathcal {C}:= \lbrace m\in \mathfrak {h}_2(\mathbb {K})|\ \hbox{\emph {det} }m = 0\rbrace \) ) with a nonnegative time coordinate \(t = |\lambda _1|^2 + |\lambda _1|^2 \ge 0\) . These two properties are important since they are the reason for the fact that supersymmetric Yang–Mills theories exists only in dimensions \((1,1+k)\)  (see e.g. [1]}, §5). Moreover this construction provides us with an orientation of the time, an important property of supersymmetric theories. Also the set \(\mathcal {C}/\mathbb {R}\)  of real lines contained in \(\mathcal {C}\) , with its canonical conformal structure, can be identified with the sphere \(S^k\)  (the heavenly sphere at infinity) or, equivalentely with the projective line \(\mathbb {K}P\) . Indeed the map \(\varphi : \mathcal {C}\setminus \lbrace 0\rbrace \longrightarrow S^k\subset \mathbb {R}\times \mathbb {K}\)  defined by \((t,x,z)\longmapsto (x/t,z/t)\)  induces a conformal diffeomorphism between \(\mathcal {C}/\mathbb {R}\)  and \(S^k\) . Note also that the image of \(X^{(\lambda _1,\lambda _2)}\)  by \(\varphi \)  is \(\left({|\lambda _1|^2 - |\lambda _1|^2\over |\lambda _1|^2 + |\lambda _1|^2}, {\lambda _1\overline{\lambda _2}\over |\lambda _1|^2 + |\lambda _1|^2}\right)\) , i.e. is nothing but the image by the Hopf fibration of \((\lambda _1,\lambda _2)\) .
